Answer:
The actual SAT score is 2024.
The equivalent ACT score is 29.49.
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean
and standard deviation
, the z-score of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
SAT score that is in the 95th percentile
Scores on the SAT test are normally distributed with a mean of 1511 and a standard deviation of 312, which means that ![\mu = 1511, \sigma = 312](https://tex.z-dn.net/?f=%5Cmu%20%3D%201511%2C%20%5Csigma%20%3D%20312)
95th percentile is X when Z has a pvalue of 0.95, so X when Z = 1.645. The score is:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![1.645 = \frac{X - 1511}{312}](https://tex.z-dn.net/?f=1.645%20%3D%20%5Cfrac%7BX%20-%201511%7D%7B312%7D)
![X - 1511 = 1.645*312](https://tex.z-dn.net/?f=X%20-%201511%20%3D%201.645%2A312)
![X = 2024](https://tex.z-dn.net/?f=X%20%3D%202024)
The actual SAT score is 2024.
Equivalent ACT score:
The equivalent ACT score is the 95th percentile of ACT scores.
Scores on the ACT test are normally distributed with a mean of 21.1 and a standard deviation of 5.1, which means that
. So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![1.645 = \frac{X - 21.1}{5.1}](https://tex.z-dn.net/?f=1.645%20%3D%20%5Cfrac%7BX%20-%2021.1%7D%7B5.1%7D)
![X - 21.1 = 1.645*5.1](https://tex.z-dn.net/?f=X%20-%2021.1%20%3D%201.645%2A5.1)
![X = 29.49](https://tex.z-dn.net/?f=X%20%3D%2029.49)
The equivalent ACT score is 29.49.